1. (a) Compute the leading digits of the first 100 powers of 2, and see how well these data fit the Benford distribution.
(b) Multiply each number in the data set of part (a) by 3, and compare the distribution of the leading digits with the Benford distribution.
2. In the Powerball lottery, contestants pick 5 different integers between 1 and 45, and in addition, pick a bonus integer from the same range (the bonus integer can equal one of the first five integers chosen). Some contestants choose the numbers themselves, and others let the computer choose the numbers. The data shown in Table are the contestant-chosen numbers in a certain state on May 3, 1996. A spike graph of the data is shown in Figure.
The goal of this problem is to check the hypothesis that the chosen numbers are uniformly distributed. To do this, compute the value v of the random variable χ2 given in Example 5.6. In the present case, this random variable has 44 degrees of freedom. One can find, in a χ2 table, the value v0 = 59.43 , which represents a number with the property that a χ2-distributed random variable takes on values that exceedv0 only 5% of the time. Does your computed value of v exceed v0? If so, you should reject the hypothesis that the contestants' choices are uniformly distributed.
|
Integer
|
Times Chosen
|
Integer
|
Times Chosen
|
Integer
|
Times Chosen
|
|
1
|
2646
|
2
|
2934
|
3
|
3352
|
|
4
|
3000
|
5
|
3357
|
6
|
2892
|
|
7
|
3657
|
8
|
3025
|
9
|
3362
|
|
10
|
2985
|
11
|
3138
|
12
|
3043
|
|
13
|
2690
|
14
|
2423
|
15
|
2556
|
|
16
|
2456
|
17
|
2479
|
18
|
2276
|
|
19
|
2304
|
20
|
1971
|
21
|
2543
|
|
22
|
2678
|
23
|
2729
|
24
|
2414
|
|
25
|
2616
|
26
|
2426
|
27
|
2381
|
|
28
|
2059
|
29
|
2039
|
30
|
2298
|
|
31
|
2081
|
32
|
1508
|
33
|
1887
|
|
34
|
1463
|
35
|
1594
|
36
|
1354
|
|
37
|
1049
|
38
|
1165
|
39
|
1248
|
|
40
|
1493
|
41
|
1322
|
42
|
1423
|
|
43
|
1207
|
44
|
1259
|
45
|
1224
|
Table: Numbers chosen by contestants in the Powerball lottery.